Ben Franklin improved the clock hundreds of years ago (and I have a modern replica to illustrate it). His assumption was that a clock could have one hand which made a rotation every 8 hours. Just as we take it for granted that we know whether it is 1 am or 1 pm (12 hour steps), he assumed that an observer can estimate the time within 8 hours. On such a clock, the minutes can be interpolated with sufficient accuracy for most human purposes.

Reminds me of a children's book I read when I was back in first grade, "Ramona the Pest". In one passage, she showed up ten minutes late for school.
Class started at quarter after eight. A quarter coin is 25 cents. Therefore, quarter past the hour must be 8:25.
It is a rather silly system, but very many of our systems have a lot of silliness in them.
Now here's one: Lat/Lon time. Instead of timezones, our time keeping devices will automatically compensate for our exact location on Earth. Since almost no one likes to talk anymore, it would be easy for our text messaging, email and any other communication system to translate so each party would see the time with their respective compensation.

I have always thought Base 16 makes more sense for everything. The Decimal System is a poorly devised system, yet everyone promotes it like it was divinely inspired.
I like Base 16 because it is divisible by 2 down to a singularity, or 1. You can't do that with Base 10. 10 divided by 2 is 5. 5 divided by 2 is 2.5 and so on. Thus we enter the realm of decimals where it get messy.
So my vote goes for 16 Hours/day, 16 minutes/hour, 16 seconds/minute. Thus a clock face would be divided into 16 parts.
Under the current system of time, when we say it's 3:15, the hour hand points to the 3, but the minute hand also points to the 3. It doesn't point to the 15.
Makes no sense to me. I think we can do better.

Mine's better. Base 2 time.
Divide the 365.256 day year into exactly 1024 time periods. I call these thirddays because they are about a third of a day long. 8.56 hours. Now we divide the third days into 8 new type hours and each hours into 64 minutes and each minute into 64 seconds.Each of these new seconds is 0.94 old seconds long. We can also now have sixteen 64 thirdday months. Now we can handle time calculations with just bit shifts.
6 bits are seconds.
6 bits are minutes.
3 bits are hours.
6 bits are days.
4 bits are months.
And the rest are years.
In practice,
Work one thirdday and take two thirddays off.
Work one thirdday and take two thirddays off.
Work one thirdday and take seven thirddays off.
Repeat.
This equates to the current 24 hour time as working 3 days out of 5 and starting work an hour and a half later everyday. For example start work at 7:00AM one day, 8:34AM the next and 10:07AM the third and then take two days off and repeat.
When do I get my patent?